21. General Directional Derivative in Terms of Gradient— Since dVids=. V V • (i cos a j cos k cos y), where the latter factor is a unit vector in direction ds, hence dVids".° NW cos 8, where 0 is the angle between VV and ds; thus the derivative of V in any direction equals the projection of V V in that direction, hence VV points in the di rection of most rapid increase of V.
22. Lines of the Gradient If a point moves continuously from one level sur face to another along successive gradients, the elements of these gradients form in the limit a continuous curve which is an orthogonal trajectory to the family of surfaces V(x, y, 2) C, and is called a line of the vector V V; and a series of such lines, passing through the perimeter of a small closed curve traced on a level surface, is called a tube of the vector.
23. Vector A vector whose three components parallel to the co-ordinate axis are given functions of (x, y, z) is called a vector function; e.g., the resultant force at the point (x, y, z) of an electric field is of the form R=iX + jY + kZ, where X, Y, Z are certain functions of (x, y, z). The simplest electric field is that due to a point charge m which repels the unit test charge placed at a distance r with a force mit"; the potential is V.-..m/r; the level surfaces are spherical; and the force F V=—dV/dr--n- sit" along r.
24. •Symbolic Scalar Product 7..a.— The . a vector •operator + + k — called the Hamiltonian operator, from its in ventor; but when used in combination it has the shorter name of del, due to Willard Gibbs; some use the name aabla suggested by Maxwell from resemblance to the Assyrian harp; Kel vin's name atled, the reverse of delta, is seldom used. The symbolic scalar product V .a a a (ia—x+ +k —) • (Jai + Jai + ka) =- ay as Oat aaz ass ax ' ay a (12), where the symbolic product of — and at is taken as &I ax ' 25. Flux of a Vector through an Area The flux of a vector a through an element of area dS is defined as the prod uct of dS by the normal component of a and is accordingly represented by a•c1S where dS is the vector of the elementary area (9). It is easy to see that the algebraic sum of the elements of outward flux through the faces of an elementary parallelopiped at (x, y, z) is . ay azi aas + —) dxdyds, which is Vat per unit volume. In case a is the velocity at (x, y, z) of a fluid v.a is the rate at which matter
is flowing out from (x, y, z) in unit time per unit volume, and is called the rarefaction or divergence at that point; and, by analogy, v.a is called the outward flux or divergence for any vector a. If a has a scalar potential, of which it is then the gradient, div V.V Vs as + + called the Laplacianof V.
ar Oza 26. Gauss' Flux Integral.— Let S be a closed surface in the field of a, divide it into small elements, and regard each element as plane, and represent it by normal vector dS drawn outwards (9), then the flux of a through S, isfia.dS; and Gauss' theorem ex presses this in terms of a volume integral over the space within S by the equation fia•dS7iff(v it)dxdyds, which is proved by summing up the flux out of the elementary parallelopipeds in (25). If a is the force due to a point charge, V a.= — V V, where V=---m/r, avox--mm(srox). — mx/r•, aiv/a0=— m —3r)/rs, V V = — m (3rt— 3x-3— 3y1-3z-i)/r 0, except where r---=0, thus except at r =0, hence by Gauss' equation, the flux through any surface not enclosing the point charge is zero, and the fluxes through all sur faces enclosing it are equal. To find this flux take a sphere of radius r with the point charge as centre, then the flux = 4ire(m/rs) If there be any number of point charges the flux of force through any surface, enclosing charges whose algebraic sum is M, is 4irM; and for a continuous distribution of density p, the outward flux is 4aff f p dxdyds. ing this tofff V (.a)dxdyds, for any enclosed volume, we have 4irp = V .a = V' V. This is Poisson's equation. At a point where p we have which is Laplace's equa tion.
27. Tubular or Solenoidal Vector Field.— Applying the flux theorem to the volume bounded by a vector tube and two normal sec tions we see, that if V.a— 0 the net outward flux is zero, hence the inward flux through one sec tion equals the outward flux through the other, since there is no flux normal to the vector lines; thus, when the tube is very thin the magnitude of a varies inversely as the normal section of the tube; hence the variation of a in magnitude and direction as we pass along a line of the vector is exhibited by the variation of the thickness and direction of the vector fila ment; and, for this reason, a vector field which has no divergence is called a tubular or solenoidal field.